PRH | Inter | 5.12 • Soft-Addition and Soft-Multiplication and the Channel–Switch Error

Soft-Addition and Soft-Multiplication and the Channel–Switch Error Blur as a Necessary Mediator Between Addition and Multiplication Aleksandar Perišić September 2025 Abstract Addition is native to the Fourier (translation) channel; multiplication is native to the Mellin (dilation) channel. After the log change of variables, the Mellin transform is a Fourier transform on the log–line. That intertwining brings along a Heisenberg–type trade–off: you can’t keep both channels perfectly sharp at once. In other words, a little blur isn’t a stylistic preference—it’s the toll you pay when you mix +and × . We package this as soft addition ⊞ and soft multiplication ⊠ , defined by inserting gentle, scale–neutral blur before we compute and only reading out at the very end. With that recipe, expressions like 3 · 2+1stop leaking hidden rounding errors: the switching is explicit, and the error budget is honest and controlled. 1 Two channels, one trade–off (in plain labels) We write M for the unitary Mellin transform (i.e., evaluated on the midline ℜs = 1 2 ), so no subscript is needed. On ( R, +), translations f ( x ) 7→ f ( x−a )diagonalize under the Fourier transform F . On (0 ,∞ )with Haar measure d x/x , dilations f ( x ) 7→ f ( ax )diagonalize under M . Setting y = log x and g ( y ) = ey/2f ( ey ), Mellin at the midline becomes a standard Fourier transform of g . The usual uncertainty then reads, with plain–English subscripts for readability, Varlog(g)·Varspec(Fg)≥1 4,(1) where Varlog is spread along the log–axis and Varspec is spectral spread. Moral: if you try to keep both addition and multiplication razor–sharp simultaneously, the math says “no”—some blur has to live somewhere. 2 A soft introduction A simple statement mixing addition and multiplication: 1+2·3 = 7. Two lenses. • Multiplicative lens (Mellin world). Numbers n are viewed through the characters ns (with s=1 2+it); multiplication is native and perfectly sharp: 2s·3s= 6s. Additive moves are nonnative here, so in general 1s+ 6s= 7s. 1 • Additive lens (Fourier/time world). Numbers n are points on the additive line; we write them as n x to emphasize the linear encoding in this lens (here x is the additive unit). Addition is native and perfectly sharp: 1x+ 6x= 7x. Multiplication is nonnative here, so a formal 2 · 3performed inside the additive lens has no sharp meaning without switching lenses. Somewhere, the switch 6s→6xmust happen. This is the story about that switch. 3 Soft blur that respects scale Ascale–neutral multiplicative blur lives on the log–line: (B× εf)(x) := Z∞ 0 Kεx uf(u)du u, Kε(r) := 1 √4πε exp−(log r)2 4ε. On the spectral side, M [ B× εf ]( t ) = e−εt2Mf ( t ), i.e., a gentle Gaussian muffler of Mellin frequencies. The additive companion gets its own one–liner, parallel in form: (B+ τf)(x) := ZR κτ(x−u)f(u) du, where κτ is an even, nonnegative approximate identity with RRκτ = 1 (e.g., the Gaussian κτ(z) = (4πτ)−1/2e−z2/(4τ)). For that Gaussian example, F[B+ τf](ξ) = e−τξ2Ff(ξ). 4 Transport operators (type-correct switching) To move cleanly between lenses, we use the unitary change of variables: (Uf)(y):=ey/2f(ey),(U−1g)(x):=x−1/2g(log x). Then F[Uf](t) = M[f](t)and M[U−1g](t) = F[g](t). Definition 1 (Soft arithmetic ⊞,⊠ ).Fix a resolution knob δ > 0and tie the channel scales via ε = ε ( δ ) , τ = τ ( δ )with ε τ ≥c > 0(the “don’t fight Heisenberg” rule). For reals x, y > 0and integers m, n ∈N, Soft multiply: x⊠y:= Round(B× εId)(xy), Soft add: m⊞n:= Round(B+ τId)(m+n). If you are working purely analytically (no discrete grid), drop Round : then ⊞,⊠ live in the blurred continuum and the final readout happens once, at the end. Remark 2 (A friendly discrete blur on N ).On the successor line, the canonical blur is the Abel profile wq ( n ) = (1 −q ) qn with q↑ 1. Think of L∼ (1 −q ) −1 as “how patient you are before you read out.” It is the natural companion of B+ τ for integer data and plays perfectly with expectations/medians. 2 5 Switching lenses without pretending A mixed expression like 3 · 2+1asks you to multiply (multiplicative lens), then nudge by +1 (additive lens). If you insist on never switching lenses, you end up faking one operation inside the other lens, and that fake introduces a quiet projection/rounding leak. Definition 3 (Named switches).We make switching explicit: Switch×→+:= B+ τ◦U◦B× ε,Switch+→× := B× ε◦ U−1◦B+ τ. Use Switch×→+ when you go from “multiply world” to “add world,” and Switch+→× for the opposite trip. Theorem 4 (No free lunch for mixing +and × ).Let B× ε, B+ τ be Gaussian (or PW) blurs with variances ε, τ. For each compact input set Kthere is CKwith  Switch×→+−Id  K≤CK·Φ(ε, τ),Φ(ε, τ)↘0only if ετ → 0. Trying to send ε↓ 0and τ↓ 0at the same time collides with (1) . So if you compute a mixed formula while refusing to switch lenses, you pick up an irreducible projection error; if you soften first and switch explicitly, the error stays in budget and goes to zero along any legal schedule. Idea in one breath. On the log side, B× ε damps Mellin frequencies by e−εt2 ; transporting via U identifies those with Fourier frequencies. On the time side, B+ τ damps Fourier frequencies by e−τξ2 . The composed switch thus has a frequency response that is the product e−(ε+τ)·(freq)2 . Heisenberg says you cannot sharpen both knobs to zero simultaneously. Quantitative bounds are a one–page Plancherel exercise. 6 Tiny demos you can feel Pure moves stay crisp 3 · 2 = 6 (multiplicative lens) and 3 + 2 = 5 (additive lens) are native, hence sharp. With soft ops, 3⊠2→6and 3⊞2→5as the single–lens variance vanishes. The mix 3·2+1 Wrong vibe (never switch): compute 6multiplicatively, then attempt to realize “+1” inside the multiply lens. Any such projection depends on a grid and leaks a rounding E: Round hadd wrong lens ,→mul(6; 1) i= 7 ±E, E = 0 in general. Notice. If you don’t pay attention, the leakage E is typically of order 1 / 4(in the normalized units induced by Varlog Varspec ≥ 1 / 4) or larger. It’s easy to mentally ignore at a single seam; but as soon as you scale the problem up, the error begins to appear in ways that (1) are impossible to control and (2) are inexplicable within a single lens. Right vibe (soften, then switch): RoundB+ τSwitch×→+[ (B× εId)(3) ·(B× εId)(2) ] + 1→7, with |error| ≤ C Φ( ε, τ )by Theorem 4. The limit is clean because we budgeted the blur up front and switched lenses openly. 3 7 Discrete side: Abel blur for integers For successor dynamics, Abel weights wq ( n ) = (1 −q ) qn with q↑ 1give the canonical “long patience” blur. Define integer–level ⊞,⊠ via expectations/medians under wq , then round once at the end. Mixed–operation error is again bounded by the same budget. 8 Budgeting, guarding, and clean limits Analytic number theory often moves data through windows protected by boundary guards and positivity in the middle; the practical way is to track a few simple “flows” (boundary fit, nonlinear remainder, moment defect, prime tail) and make them summable. Our story is the same: blur first ( B+ τ, B× ε ), switch explicitly ( Switch×→+,Switch+→× ), keep an eye on the budget Φ(ε, τ), and read out once. 9 One–page recipe 1. Do native steps in their native lens (products in multiply; increments in add). 2. Insert gentle, scale–neutral blur in each lens (B× ε, B+ τ). 3. When you must mix, switch lenses explicitly with Switch×→+or Switch+→×. 4. Schedule the scales so that Φ(ε, τ)→0without violating ετ ≳1. 5. Round only once, at the very end. 10 Euler’s constant as a synchronized finite part of two blurs This section isolates the precise role of Euler’s constant γ= lim x→∞ X n≤x 1 n−log x, as the synchronized finite part that remains when we (i) pass from an infinite object to a finite/smoothed one (truncation/regularization blur), and (ii) switch lenses between multiplication and addition (the + ↔× blur). In our language: there are two uncertainty knobs, and γ is the constant that survives when both are sent to zero along any honest schedule. Two channels and their mandatory blur Addition lives on ( R, dx )with Fourier analysis; multiplication lives on ((0 ,∞ ) , dx/x )with Mellin analysis. The map x = eu converts multiplicative structure into additive structure on the log-line u = log x . Any computation that is performed in one channel and read in the other must pay a small blur—concretely, a smoothing on the log-line before switching lenses. We formalize this with a (multiplicative) blur kernel κτon the log-line: κτ∈C∞ c(R), κτ≥0,ZR κτ(u)du = 1, κτ−−→ τ↓0δ0, and write f∗κτfor convolution in u(equivalently, multiplicative smoothing in x). 4 Three faces of γ(two knobs at once) •Discrete →continuous (finite vs. infinite). The classical limit X n≤x 1 n= log x+γ+o(1) (x→ ∞) exhibits γas the finite-part constant left after removing the main trend log x. •Prime side, additive form (Mertens). Taking logs of Mertens’ product Y p≤x1−1 p∼e−γ log x yields the additive equivalence γ+ log log x+X p≤x log1−1 p−→ 0,(2) i.e. two observations of the same object at different “wavelengths” (additive log log x vs. multiplicative prime product) are reconciled by the same constant γ. •Archimedean switch cost (digamma). Near s= 1 we have ζ(s) = 1 s−1+γ+O(s−1), ψ(s) = Γ′(s) Γ(s), ψ(1) = −γ. The digamma value ψ (1) is the archimedean (real-place) normalization that appears whenever we match Mellin-side data (multiplicative) to Fourier-side data (additive). It packages the +↔× blur into a single constant. Kernel-invariant extraction of γvia multiplicative blur Define the blurred harmonic profile on the log-line by Hτ(u) := X n≥1 1 nκτu−log n. Lemma 5 (“Blur then switch”).For any approximate identity κτas above, Hτ(u) = u+γ+o(1) (τ↓0), with the o(1) uniform for uin compact sets. Proof sketch. Take the Fourier transform in u: c Hτ(t) = c κτ(t)X n≥1 1 ne−it log n=c κτ(t)ζ(1 + it). As τ↓0,c κτ(t)→1pointwise and is uniformly bounded. Using the expansion ζ(1+it) = 1 it +γ+O(t) (t→0), we invert the transform: the term 1 / ( it )yields the ramp u , while the finite part γ yields the constant intercept. Standard Tauberian/Paley–Wiener bounds justify passing to the limit under the inverse transform, giving Hτ(u) = u+γ+o(1). Remark 6 (Two-knob interpretation).Lemma 5exhibits γ as the kernel-invariant intercept that remains after simultaneously sending (i) the truncation blur (finite vs. infinite) and (ii) the channel-switch blur (+ ↔× ) to zero. Changing the blur shape changes intermediate terms but not the limit. 5 Prime-side additive statement The line γ+ log log x+X p≤x log1−1 p−→ 0 is precisely the log of Mertens’ product and is the prime-side mirror of the harmonic-sum statement: the difference between the additive observable log log x and the multiplicative observable Pp≤xlog (1 − 1 /p )isapure constant in the limit, namely −γ . From the “wave vs. particle” viewpoint: log log x measures density at additive wavelength, the prime product measures multiplicative cohesion; γsynchronizes the two. A tidy split: finite-part +archimedean Fix a legal soft-switch scheme (blur kernel and normalization). Then one can write γ=γfinite |{z} discrete→continuous +γarch |{z} +↔× (digamma at 1) ,0≤γarch ≤ −ψ(1) = γ (equivalently, 0 ≤γfinite ≤γ ). The two summands individually depend on the adopted scheme, but their sum is invariant. Operationally, γfinite is the finite-part constant in Pn≤x1 n−log x , while γarch is the switch-cost enforced by the Γ-factor at the real place. Any change in the blur shifts these two pieces by equal and opposite amounts, leaving γunchanged. Drop-in recipe (“blur →switch →read”) To compute the intercept consistently across the two channels: 1. Choose a log-line mollifier κτwith Rκτ= 1. 2. Form Hτ(u) = Pn≥11 nκτ(u−log n). 3. Read in the additive channel: Hτ(u)=u+γ+o(1) as τ↓0. 4. On the prime side, use (2) to see the same constant by comparing log log x to Pp≤xlog (1 − 1/p). Remark 7 (Practical moral).The constant γ is not an accident of a specific truncation; it is the blur-invariant that guarantees that additive and multiplicative observables of the same phenomenon match up to a universal finite part. In our framework, it is the synchronized limit of the two uncertainty knobs. 11 Why the soft–switch discipline matters Mixing additive and multiplicative structure without softening and explicit switching does not make the error disappear; it only hides it. In prime problems those hidden leaks become first–order obstacles. Here are the places where the discipline “blur, then switch, then read out once” is not optional: • Explicit–formula / Perron cutoffs. A hard indicator 1 n≤x is an additive knife, while the Dirichlet series A ( s ) = Pa ( n ) n−s and its Perron integral live on the multiplicative side. Replacing 1 n≤x by a bump ψ ( n/x )with ψ∈C∞ c is exactly inserting B+ τ before crossing to Mellin. On the Mellin side, M [ ψ ( · )]( t )adds a benign spectral muffler; without it, boundary terms (from an abrupt switch) dominate the error. In our language: apply 6 B+ τ , use Switch+→× to cross into the multiplicative lens, run the argument, and only then round. X n≤x a(n)⇝X n≥1 a(n)ψn x=1 2πZRM[ψ](t)A1 2+itx1 2+it dt. • Short intervals and correlations. For S ( x, H ) = Px<n≤x+H Λ( n )or pair counts P Λ( n )Λ( n + h ), the window is additive, while multiplicative tools (Dirichlet series, Euler products) are used to analyze. The soft window ψn−x H ( B+ τ at scale H ) localizes Fourier frequencies and keeps the switch budget Φ( ε, τ )under control when you move through Switch+→× and back. • Multiplicative f ( n )vs. additive phases e ( αn )(circle method). Major/minor–arc decompositions are literally “switch, then localize.” Pre–blurring the additive phase isolates rational arcs cleanly; skipping it fakes an additive tweak inside the multiplicative lens and quietly injects projection/rounding error into the “minor–arc cancellation” you claim. • Dirichlet convolution vs. ordinary convolution. Dirichlet convolution ( f∗g )( n ) diagonalizes under M ; ordinary convolution ( u∗v )( x )diagonalizes under F . Treating one as the other without Switch×→+/Switch+→× introduces aliasing: mean–value identities acquire spurious cross–terms that do not vanish unless a blur was present. • Mollifiers and pretentious distances. A mollifier is a codified B× ε : it smooths the multiplicative spectrum before correlating primes with L –function zeros. Without that softening, the same integrals either blow up in constants or lose the “good” lower bounds entirely. The soft–switch viewpoint makes those choices structural, not ad hoc. • Tauberian steps (partial summation as a switch). Passing from Pa ( n ) n−s to Pn≤xa ( n )needs a soft transition (Abel/partial summation kernel). This is nothing but B+ τ composed with Switch+→× . Doing it hard is the classic “forgot to switch lenses” mistake. Five–rule checklist (use every time). 1. Tag the native lens. Products, divisibility, Dirichlet convolution → multiplicative; shifts, increments, short windows →additive. 2. Blur before you mix. Insert B× εon the log–side, B+ τon the time–side. 3. Switch explicitly. Use Switch×→+ or Switch+→× ; never fake an additive move inside the multiplicative lens (or vice versa). 4. Respect the scale constraint. Keep ε τ ≳ 1(Heisenberg). Drive the budget Φ( ε, τ ) → 0 only along legal schedules. 5. Round once, at the end. Intermediate rounding = hidden aliasing. With this discipline the “mysterious” failures in prime–feature arguments (spiky boundary terms, unstable constants, phantom oscillations) simply do not appear: the limits are clean because the switches were honest and the blur was budgeted up front. 12 Closing note Blur is not a hack; it’s good manners between two lenses that don’t speak the same native tongue. If you admit a tiny softness where it belongs, the final limit is cleaner because the process is honest about what the uncertainty allows—and you never have to hide a projection under the rug. 7